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Software Verification 1 Deductive Verification. Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität und Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik. Lehrevaluation. Verpflichtend für die HU, im Interesse der Studierenden
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Software Verification 1Deductive Verification Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität und Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik
Lehrevaluation • Verpflichtend für die HU, im Interesse der Studierenden • Zeitraum: 16.01. bis 27.01.2012 • online: https://evaluation.hu-berlin.de/evaluation/ • Passwort (Token): inf-ws-11-12 • Verbesserung der Sicherheit durch sogenanntes Captcha • Completely Automated Public Turing test to tell Computers and Humans Apart • Bei Rückfragen: Dr. Elke Warmuth, Studiendekanin • Tel. 2093 5830, E-Mail: warmuth@math.hu-berlin.de
Pre- and Postconditions • Dijkstra: wp-calculus (weakest precondition) • characterize the “weakest” formula which makes a Hoare-triple valid • =wp(.) iff ⊢ and⊢(') for every ’ for which ⊢’ • =wlp(.) iff ⊢{}{} and⊢(') for every ’ for which ⊢{’} {}(weakest liberal precondition, see later) • Example: wp(x++, x==7) = (x==6) • Dijkstra gives a set of rules for wp which can be seen as notational variant of Hoare logic
wp(skip, ) = • wp(x=t, ) = [x:=t] • wp({1; 2}, ) = wp(1, wp(2, )) • wp(if (b) 1 else 2, ) =((b wp(1, )) (¬b wp(2, ))) • wp(while (b) , ) = z (z) z((b(z)) z’ (z’<z wp(, (z’))) z((¬b(z)) )where is a loop variant and < a wfo, z new var. ! This is a non-constructive definition ! Existence???
Examples • wp(x=x-3, x>7) = x>7 [x:=x-3] = x-3>7 = x>10 • wp({x*=2; x-=3}, x>7) = wp(x*=2, wp(x-=3, x>7)) = wp(x*=2, x>10) = x>5 • wp(if(a<b) a=b, a>=b) = ((a<b wp(a=b, a>=b) (a>=b wp(skip, a>=b))=((a<b b>=b) (a>=b a>=b)) = T • wp(while (i>0) i--, i==0) = i>=0
Partial Correctness • Weakest liberal precondition wlp(,) • wlp(while (b) , ) = ((b) wlp(, )) ((¬b) ) • Dijkstra also used nondeterministic programs („guarded commands“) • guarded-command-program ::= while-program | guarded-command • guarded-command ::= b : e | b : e [] guarded-command • b: condition, e: guarded-command-program
Strongest Postconditions • Dual to weakest precondition: the strongest formula which can be guaranteed to hold after execution • =sp(, ) iff ⊢ and⊢( ') for every ’ for which ⊢ ’ • sp(x=t, )= z (x==t[x:=z] [x:=z]) (z new) • e.g. sp(x=x-3, x>7) = z (x==z-3 z>7) = x>4 • Pre- and postconditions are important in the presence of methods and procedures
Functions and Procedures • while-Programs: • whileProg ::= skip | V=T| {whileProg;whileProg} | if (FOL-) whileProg else whileProg | while (FOL-) whileProg • T is the set of terms in the signature =(D, F, R) • Now: extended signature ’=(D{void}, FF’,R) • If f is of type void, then f(x1,...xn) is an (imperative) program • term ::= F(T, ..., T) | F’(T, ..., T) • for each f F’ there must be a declaration: • decl ::= type F’ (V, ... V); whileProg • V in decl are called formal parameters • T in terms are called actual parameters
No alias: formal parameters should be pairwise different • No scoping: formal parameters must be different from program variables • return statement as assignment to the function name • If a function or procedure name occurs directly or indirectly in the call graph of its declaration, it is called recursive • for the time being: no recursion • There are various ways to pass actual parameters for formal ones (value, reference, name, ...) • for the time being, we use only call-by-value • passing value w to formal parameter v has the same effect as the assignment v=w at the entry of the procedure or function
Example intgcd(int a, int b) • while (a!=b) { c = max(a,b)-min(a,b); a = min(a,b); b = c; • } } int min (int a, int b) if (a<b) min=a else min=b; int max (int a, int b) if (a>b) max=a else max=b;
Example int min (int a, int b) if (a<b) min=a else min=b; {x = 5; y = 7; z = min (x, y)} is equivalent to { x = 5; y = 7; a = x; b = y; if (a<b) min=a else min=b; z = min; } need pre- and postconditions to show assertions.
Example intgcd(int a, int b) • {a==m>0 b==n>0} while (a!=b) { c = max(a,b)-min(a,b); a = min(a,b); b = c; • } • gcd = a; • {gcd|m gcd|n ...} } int min (int a, int b) if (a<b) min=a else min=b; {a<=min b<=min (a=min b=min)} int max (int a, int b) if (a>b) max=a else max=b; {a>=max b>=max (a=min b=min)}
Contracts • weakest preconditions and strongest postconditions are related to the require-ensure-paradigm (also called assume-guarantee-paradigm): /*@ requires ensures */void foo(...) ; is equivalent to (wp(,)) (sp(, )) • such a statement is called contract • use of contract: {[x1:=t1, ..., xn:=tn]} foo(t1,...,tn) {}
